This tool sums n identical normalized |sin| waveforms separated by a selected phase: S(θ) = Σ|sin(θ + kφ)|. It is a mathematical phasing demonstrator, not an engine torque model and not a multicylinder valve-event, balance, self-start or flywheel calculator.
Idealized Rectified-Sine Phase Sum
Explore how identical normalized |sin| waveforms combine when separated by a selected phase.
Equation Used
- Every component uses the same unit-amplitude absolute-sine waveform.
- Components are separated by the selected constant phase.
- The current evaluation angle is independent of the animation scrubber.
- Pressure, valve events, crank-slider geometry, inertia, friction and loads are excluded.
Minimum, maximum and mean are sampled at 3,600 equally spaced positions over one mathematical cycle.
Selected waveform model
Choose one to four component waveforms, an angular separation φ and evaluation angle θ. Each component contributes |sin(θ + kφ)| and the tool displays their sum.
The absolute-sine assumption makes every contribution nonnegative and equal in amplitude. A real cylinder does not obey that rule through a full revolution; its contribution depends on pressure, valve events, acting arrangement, crank-slider geometry and inertia.
What the demonstrator reports
The outputs are the current normalized sum, its sampled minimum over one revolution, the minimum-to-maximum ratio and peak-to-peak range divided by the sampled mean.
These describe only the constructed waveform. They do not establish engine torque, self-start capability, power, balance, bearing load, valve timing or suitable crank phasing.
Idealized phase sum
S(θ) = Σ from k = 0 to n − 1 of |sin(θ + kφ)|. The current value is evaluated at the selected θ. Minimum, maximum and mean are sampled at 3,600 equally spaced positions over 0 to 2π.
The displayed min/max percentage is 100 Smin/Smax. The displayed variation is 100(Smax − Smin)/S̄. These are mathematical waveform metrics, not measured torque ripple.
Selected-input example
For n = 2, φ = 90° and θ = 0°, the two contributions are 0 and 1, so the normalized sum is 1. The sampled minimum over the constructed cycle is also 1.
This illustrates complementary phase placement in the chosen functions. It does not prove that a two-cylinder engine produces that torque curve or starts from every position.
What the engine animation shows
Hiscox describes three or more single-acting cylinders attached to a revolving flywheel. Their connecting rods act against a stationary crank set off-center from the flywheel axis. Valves lie on a central disc. The animation shows three cylinders, their pistons and the fixed crank with illustrative dimensions.
The piston motion comes from constant-length connecting rods and the eccentric crank position. The waveform calculator is a separate mathematical exercise: its component count and phase separation do not set the illustrated engine layout. Predicting engine torque would also require pressure diagrams, valve timing, friction, inertia and load data.
Questions
Why use absolute sine?
It creates a simple nonnegative periodic waveform for visualizing phase overlap. It is a chosen abstraction, not a cylinder-pressure law.
Does a positive minimum prove self-starting?
No. Starting requires actual pressure and valve events, direction, static friction, load torque and geometry at the stopped position.
Does the phase setting recommend crank angles?
No. It only offsets identical mathematical waveforms. Engine crank arrangement depends on the specified architecture and dynamics.
Is the displayed variation actual torque ripple?
No. It is peak-to-peak variation of the normalized phase sum divided by its mean.
Technical references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), page 95, item 350: rotary multi-cylinder engine.
- OpenStax Physics, Angle of Rotation and Angular Velocity — angular position and phase context.
- OpenStax Precalculus, Unit-Circle Sine and Cosine Functions — the periodic functions used in the constructed phase sum.
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